parametric smoothness and self-scaling of the statistical...

Physica D 234 (2007) 105–123www.elsevier.com/locate/physd

Parametric smoothness and self-scaling of the statistical properties of aminimal climate model: What beyond the mean field theories?

Valerio Lucarinia,b,∗, Antonio Speranzaa, Renato Vitoloa

a PASEF – Physics and Applied Statistics of Earth Fluids, Dipartimento di Matematica ed Informatica, Universita di Camerino, Camerino (MC), Italyb Department of Physics, University of Bologna, Bologna, Italy

Received 9 October 2006; received in revised form 2 July 2007; accepted 8 July 2007Available online 14 July 2007

Communicated by U. Frisch

Abstract

A quasi-geostrophic intermediate complexity model of the mid-latitude atmospheric circulation is considered, featuring simplified baroclinicconversion and barotropic convergence processes. The model undergoes baroclinic forcing towards a given latitudinal temperature profilecontrolled by the forced equator-to-pole temperature difference TE . As TE increases, a transition takes place from a stationary regime – Hadleyequilibrium – to a periodic regime, and eventually to a chaotic regime where evolution takes place on a strange attractor. The attractor dimension,metric entropy, and bounding box volume in phase space have a smooth dependence on TE , which results in power-law scaling properties. Power-law scalings with respect to TE are detected also for the statistical properties of global physical observables — the total energy of the system andthe averaged zonal wind. The scaling laws, which constitute the main novel result of the present work, can be thought to result from the presenceof a statistical process of baroclinic adjustment, which tends to decrease the equator-to-pole temperature difference and determines the propertiesof the attractor of the system. The self-similarity could be of great help in setting up a theory for the overall statistical properties of the generalcirculation of the atmosphere and in guiding – on a heuristic basis – both data analysis and realistic simulations, going beyond the unsatisfactorymean field theories and brute force approaches. A leading example for this would be the possibility of estimating the sensitivity of the output ofthe system with respect to changes in the parameters.c© 2007 Elsevier B.V. All rights reserved.

Keywords: Atmospheric circulation; Climate model; Chaotic dynamics; Strange attractor; Attractor dimension; Metric entropy; Bonding box; Scaling law; Self-similarity

1. Introduction

The understanding of climate is a scientific issue as wellas being also of practical importance, since it is connected toaspects of vital importance for human life and environment.Recently, climate has also become a politically relevantissue. In this paper we propose our scientific approach andsome encouraging results concerning the theory of GeneralAtmospheric Circulation (GAC) – the core machine of weather

∗ Corresponding address: Department of Mathematics and ComputerScience, University of Camerino, Via Madonna delle Carceri, 62032 Camerino,MC, Italy. Tel.: +39 3476141563.

E-mail addresses: [email protected], [email protected](V. Lucarini).

0167-2789/$ - see front matter c© 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2007.07.006

and climate – looked upon as a problem in physics. Wenote here that there may be evidence that our understandingof the dynamical processes determining climate, and itsevolution, is somewhat lagging behind our ability in setting up(often somewhat gigantic) simulation models of the climaticsystem [53] and, as a consequence, from the proposed effortgreat benefits can come to our ability to diagnose and/orconstruct climate models.

1.1. The climatic system

Climate is defined as the set of the statistical properties of theclimatic system. In its most complete definition, the climaticsystem is composed of four intimately interconnected sub-systems, atmosphere, hydrosphere, cryosphere, and biosphere.

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106 V. Lucarini et al. / Physica D 234 (2007) 105–123

These subsystems interact nonlinearly with each other onvarious time–space scales [51,59]. The atmosphere is themost rapid component of the climatic system, is very rich inmicrophysical structure and composition, and evolves under theaction of macroscopic driving and modulating agents — solarheating and Earth’s rotation and gravitation, respectively. Theatmospheric circulation is the basic engine which transformssolar heating into the energy of the atmospheric motions anddetermines the bulk of weather and climate as we commonlyperceive them. The atmosphere features both many degreesof freedom, making it complicated, and nonlinear interactionsof several different components involving a vast range oftime scales, which makes it complex. The dynamics of sucha system is chaotic and is characterized by a very widespectrum of natural (i.e. internally generated) variability [48].The understanding of the physical mechanisms operatingin the atmosphere critically influences important humanactivities like weather forecasting, territorial planning, etc.This is one of the reasons why von Neumann positionedatmospheric dynamics in the core of the ongoing developmentof numerical modeling [14]. The GAC also poses problemsof a general physical nature as a realization of planetaryscale thermodynamic transformations in a rotating, stratifiedfluid. Among all the physical processes involved in the GAC,the baroclinic conversion plays a central role because it isthrough this mechanism that rotating, stratified fluids convertthe available potential energy [45], stored in the form of thermalfluctuations, into the vorticity and kinetic energy of the airflows. At mid-latitudes of both hemispheres, the baroclinicconversion process can be taken as the main process responsiblefor the destabilization of any hypothetical stationary circulation(fixed point, in the terminology of dynamical systemstheory) of the atmosphere, in particular that given by thezonally (longitudinally) symmetric atmospheric circulationcharacterized by a purely zonal (west to east) wind (jet) [36].Note that these theoretically computed stationary circulationscorrespond to values of the field variables way out of theobserved range: the zonal solution is characterized by windsof the order of 100 ms−1 and the latitudinal (equator–pole)temperature difference to the order of 100 K. Both of thesevalues are about twice as large as what actually observed. Apartfrom the classical role of baroclinic instability [12,19] in mid-latitude weather development [21,22,76], different forms ofbaroclinic conversion can be actually observed, both at largerscales (e.g. in the ultralong planetary waves associated with theso called low frequency variability [5,6,15,68]) and at smallerscales (e.g. in the banded sub-frontal structures [55]). Thedefinition of the basic ingredients in the physical mechanismof baroclinic conversion is to be considered one the mainsuccesses of the dynamical meteorology of the past century.

1.2. Classical theories of GAC and associated problems

Historically – see the classical monograph by Lorenz [47]– the problem of GAC has been essentially approached interms of analyzing the time-mean circulation. Usual separationssuch as between time average and fluctuations or between the

zonal and the eddy field, are somewhat arbitrary, but perfectlyjustifiable with arguments of symmetry, evaluation of statisticalmoments, etc. The time average–fluctuations separation is oftensuggestive of a theory in which the fluctuations growing on anunstable basic state, identified in the time average, feed backonto the basic state itself and stabilize it. The central objectiveof the classical circulation theory is the closure in the formof a parameterization of the nonlinear eddy fluxes in termsof quantities which can be derived from the mean field itself.Extremely important and interesting theories of this kind – e.g.baroclinic instability – have been formulated throughout thewhole history of meteorology and have substantially improvedour understanding of the basic dynamical processes underlyingthe observed evolution of the atmospheric system. However,in general this approach is not only unsuccessful, it is justnot feasible. The basic reason is that the stability propertiesof the time-mean state do not provide even a zeroth-orderapproximation of the dynamical properties of the full nonlinearsystem. This has been illustrated by theoretical arguments [24]and simple counter-examples of physical significance (seee.g., [54,78]). As a consequence, it is not possible to createa self-consistent theory of the time-mean circulation relyingonly on the time-mean fields, since a parameterization of eddyfluxes in terms of quantities derivable from the mean field isuseless: the average state does not carry the necessary physicalinformation.

Alternative approaches have been proposed, e.g. mimickingthe GAC via dynamical systems approach with simplified lowdimensional models, say below 10 degrees of freedom (seee.g. [34]). However, the physical relevance and mathematicalgenerality of these approaches has been criticized aswell, whereas the relevance of the space–time geophysicalscaling behavior has been emphasized [69]. Nevertheless, acomprehensive and coherent picture of the GAC is still far frombeing available and the classical time-mean approach has notbeen completely abandoned as a paradigm.

1.3. Some consequences of the failure of classical GACtheories

It is clear that in the context of the classic paradigm ofGAC, the main task of modeling consists in capturing the meanfield: the fluctuations will follow as its instabilities. This pointof view, whether explicit or not, has strongly influenced theset-up and diagnostics of existing General Circulation Models(GCMs). As a consequence, the diagnostic studies usually focuson comparing the temporal averages of the simulated fieldsrather than on analyzing the representation of the dynamicalprocesses provided by the models. Similar issues are related tothe provision of the so-called extended range weather forecasts.Suppose, in fact, that the forecaster was given the next monthaverage atmospheric fields: what practical information couldhe/she derive from that? Of course, if dynamical informationwere stored in the average fields – typically in the formof dominant regimes of instability derivable from the time-mean flow – he/she could obtain useful information from theprediction of such time-mean fields. Unfortunately, as remarked

V. Lucarini et al. / Physica D 234 (2007) 105–123 107

above, such a picture has proved to be far from being applicable,and the problem of extended range forecast is still open even interms of clearly formulating what we should forecast in placeof average fields.

1.4. The statistical mechanical point of view

Conspicuous difficulties were encountered [3,44] whiletrying to construct a Climate Change theory on the basisof fluctuation–dissipation theorems [42]. The main problemis that the climate system may be considered as a forcedand dissipative chaotic system, featuring properties suchas the non-equivalence between the external and internalfluctuations [49]. The reason for this non-equivalence is that,by dissipation, the attractor of the system lives on a manifoldof zero volume inside phase space [20] (although a smallamount of noise, which is always present in physical systems,smooths the invariant measure corresponding to the attractor[65]): the internal natural fluctuations occur within such amanifold whereas externally induced fluctuations move thesystem out of the attractor with probability one. In thissense, generalizations of the fluctuation–dissipation theoremhave been recently established for certain systems having achaotic attractor. Ruelle [67] has proved a generalization ofthe fluctuation–dissipation theorem for Axiom-A attractors(such systems carry a unique, zero-noise SRB measure [81]).Furthermore, some specific examples have been providedin [11] and a further extension to finite perturbations has beenrecently obtained [8]. At numerical level, Kramers–Kronigrelations [52] have been recently verified [62] also for thelinear response of the Lorenz-63 system [46], which is non-hyperbolic. Nevertheless, to the best knowledge of the authors,rigorous applications of these results to the case of GAC havenever been performed.

A naturally ensuing scientific goal is to derive simplifiedequations of motion (minimal models). The project of writingthe equations describing the statistics of the system directly onthe attractor (by projection) has been carried on with greatattention in the recent past, but, despite the big efforts, noapplicable result has been obtained so far and basic difficultieshave emerged [28,31,50].

1.5. The brute force approach

The public attention on Climate Change issues has motivatedin recent years an enormous increase in the number of degreesof freedom (up to 106 and beyond) of the most recent GlobalCirculation Models (GCMs) and in the complexity of thephysical processes included in them [39]. In the absence ofrobust and efficient scientific paradigms, a gap was createdbetween the studies revolving around phenomenological and/ornumerical modeling issues and those focused on fundamentalmechanisms of GAC [35]. Note that the assumption thatadopting models of ever increasing resolution (allowed by thelarger and larger availability of computer power) will eventuallylead to the final understanding of the GAC (a sort of bruteforce approach) is not based on any consolidated mathematical

knowledge and may, in fact, be misleading. One basic reasonis that, in the limit of infinite resolution for any numericalmodel of fluid flow, the convergence to the statistical propertiesof the continuum real fluid flow dynamics is not guaranteed.At the present stage, most of the leading climate models areneither consistent nor realistic even in the representation of thebasic time–space spectral properties of the variability of theatmosphere at mid-latitudes [53].

1.6. Numerical modeling, smoothness and limited variability

In the modeling activity we have to deal with three differentkinds of attractors: the attractor of the real atmosphere; thereconstruction of the real attractor from observations; theattractors of the model (maybe more than one) we adopt inclimatic studies. As for the second one, the (presumably)best reconstructions available are provided by the reanalyses.Reanalyses are obtained by means of variational adaptation(in simple quadratic error model–observations measure) of aclimate model trajectory to observations. Note that many ofthe operations that are practically performed when executingreanalysis (e.g., local linearizations in phase space) requireproperties of local smoothness in phase space. Presentlythere are two alternative main global reanalyses, whichfeature distinct statistical properties, albeit obviously in broadagreement. Details can be found, e.g., in [21,22,68]. As for thethird kind of attractors, the various GCMs almost surely possesssubstantially different attractors, which reflects disagreementsin the statistical properties of the atmosphere [53]. Inthe fundamental effort of bringing a model into statisticalagreement with the reanalyses (taken as best guess of the realatmosphere), a key role is played by the smoothness and thelimited variability of the statistical properties of the model asa function of the tunable parameters. Note that if the statisticalproperties of a model feature a too strong dependence on thetunable parameters, the shooting to the target of the statisticalproperties of reanalyses would be prohibitively difficult and ill-defined.

Actually, the properties of low dimensional prototypes ofatmospheric circulation are rather discouraging in this respect(see e.g. [9]), since small changes in the values of the modelparameters may cause dramatic variations of the resultingclimate. However, a sort of common wisdom among modelershas always been that with the increase in the number of degreesof freedom the statistical properties could somewhat regularizethe discontinuous dependence on external parameters thatcharacterizes many small dimensionality models. See e.g. [1,2].

1.7. Our approach

Our approach focuses on the following question: once weabandon the unsatisfactory mean field theories, what maycome next? More specifically, what can we do for developinga new approach to a theory of GAC which can serve asa guidance for setting up and diagnosing GCMs? Whichstatistical properties may be crucial? In so doing, we are, atthis stage, not seeking realism in the model representation of


atmospheric dynamics, but rather searching for representationsof key nonlinear processes with the minimum of ingredientsnecessary to identify the properties under investigation. Wehave decided to choose a rather simplified quasi-geostrophic(QG) model with just a few hundreds degrees of freedom, ableto capture the central process of the GAC, i.e. the ordinarybaroclinic conversion in the mid-latitude atmospheric jet. Otheratmospheric processes, acting on longer or shorter spatial andtemporal scales, such as those mentioned before, are essentiallyexcluded. The model is vertically discretized into two layers,which is the minimum for baroclinic conversion to takeplace [58,60], and latitudinally discretized by a Fourier half-sine pseudospectral expansion up to order JT . We have usedJT = 8, 16, 32, 64, yielding a hierarchy of QG models havingincreasing phase space dimension. A fundamental property ofthese models is almost-linearity: the eddy field is truncated toone wavenumber in the longitudinal (zonal) direction, whichentails the evolution equation of the waves to be linear interms of the time-varying zonal flow. This provides a dynamicalmeaning for the separation between zonal and eddy flow,where the zonal wind acts effectively as an integrator. Themodel considered here has extensively been used in nonlinearmodeling analysis since the ’80s [54,78] and, more recently,has been adopted as a dynamical simulator for analysis ofextremes [26,27]. Of course, today there is a vast choice ofreadily available analogous or more realistic models, whichcan be run with ease with easily available computer facilities,but our choice of intermediacy between very low dimensionalmodels and GCMs allows for a detailed check of all thestatistical properties of the model dynamics.

1.8. The structure of the paper

In Section 2 we present a brief derivation of the evolutionequations for the two-layer QG model. This derivation allowsa clear understanding of the physics involved in the consideredhierarchy of QG equations. The main results of this work arepresented in Sections 3 and 4. We study the sensitivity of themodel behavior with respect to the parameter TE determiningthe forced equator-to-pole temperature gradient, which actsas baroclinic forcing. The influence of the order of (spectral)discretization in the latitudinal direction is also analyzed. InSection 3 we characterize the transition from stationary tochaotic dynamics and study the dependence on TE and onmodel resolution JT of the dimension of the strange attractor,of the metric entropy, and of the volume of its bounding boxin the phase space. In Section 4 we analyze the statisticalproperties of two physically meaningful observables (functionsof state space variables), namely the total energy of the systemand the latitudinally averaged zonal wind. An inspection ofthe latitudinal wind profiles is also presented. In Section 5 adiscussion is given of the properties observed for the presentmodel in terms of dynamical systems theory and of the relevantphysical implications. In Section 6 we give our conclusiveremarks and perspectives for future works. In the Appendix wepresent the set of differential equations examined in this studyand sketch the numerical methods adopted.

Fig. 1. Sketch of the actual geographical area corresponding to the simplifiedβ channel. The local x and y directions and the β-channel width L y areindicated. The mid-latitudes range from 1/4L y to 3/4L y , corresponding to a45◦ latitudinal belt centered at 45◦ N.

2. The model

The description of the large scale behavior of the atmosphereis usually based on the systematic use of dominant balances,which are derived on a phenomenological diagnostic basis, butwhose full correctness at theoretical level is still unclear. Whenconsidering the dynamics of the atmosphere at mid-latitudes,on spatial and temporal scales comparable with or larger thanthose of the synoptic weather (about 1000 km and 1 day,respectively), the hydrostatic and geostrophic balances arephenomenologically well established. From the set of ab-initiodynamic and thermodynamic equations of the atmosphere it ispossible to obtain a set of simplified prognostic equations forthe synoptic weather atmospheric fields in a domain centered atmid-latitudes – the QG equations – by assuming that the fluidobeys the hydrostatic balance and undergoes small departuresfrom the geostrophic balance [13,58,77]. A great number ofphysical phenomena are filtered out of the equations by theQG approximation: various types of waves associated withstrong local divergence, turbulent motions, etc. Again, thereis no doubt that these are small on the time–space scales ofthe motions we consider. However, it is still an open questionto what extent they influence or not the statistics of largescale atmospheric motions and, in case, how to model such astatistical effect. Note also that in general the QG attractor isnot a good approximation to the attractor of the correspondingfull ab-initio equations, despite the fact that the QG balanceapproximation is diagnostically quite good for the dominatingtime–space scales of the atmosphere [50].

In this work we consider a β-channel periodic domain,with ∈ R/2πLx denoting the zonal and y ∈

[0, L y

]the

latitudinal coordinate (see Fig. 1). As a further approximation,


Fig. 2. Left: Sketch of the vertical–longitudinal section of the system domain. The domain is periodic in the zonal direction x with wavelength Lx . At each pressurelevel, the relevant variables are indicated. Right: Number of linearly unstable modes at the Hadley equilibrium as a function of the parameter TE for JT = 8, 16, 32,64.

we consider only two vertical layers [60]. This is the minimalsystem retaining the baroclinic conversion process, which is thebasic physical feature of the QG approximation. We refer to theleft panel of Fig. 2 for a sketch of the vertical geometry of thetwo-layer system. In order to avoid problems in the definition ofthe boundary conditions of the model, due to the prescription ofthe interaction with the polar and the equatorial circulations atthe northern and southern boundary, respectively, we considera domain extending from the pole to the equator. Of course,near the equator the QG approximation is not valid, so that therepresentation of the actual tropical circulation is beyond thescope. The equations of motions are:

D1t (∆Hψ1 + f0 + βy)− f0

ω2 − ω0

δp= 0, (1)

D3t (∆Hψ3 + f0 + βy)− f0

ω4 − ω2

δp= 0, (2)

D2t

(ψ1 − ψ3

δp

)− H2

2f0

p22ω2

= κ∆H

(ψ1 − ψ3

δp

)+

Rp2 f0

Q2

C p(3)

where f0 is the Coriolis parameter and β its meridionalderivative evaluated at the center of the channel, κ

parameterizes the heat diffusion, R and C p are thethermodynamical constants for dry air, ∆H is the horizontallaplacian operator. Moreover, the streamfunction ψ j is definedat pressure levels p = p j=1 = p0/4 and p = p j=3 = 3/4p0,while the vertical velocity ω is defined at the pressure levelsp = p j=0 = 0 (top boundary), p = p j=2 = p0/2, andp = p j=4 = p0 (surface boundary). The pressure levelpertaining to the discrete approximation to vertical derivative ofthe streamfunction ∂ψg/∂p as well as the stratification heightH is p = p j=2, and δp = p3 − p1 = p2 = p0/2. Q2 isthe diabatic heating, and D j

t is the usual Lagrangian derivativedefined at the pressure level p j , D j

t • = ∂t + J(ψ j , •

), where

J is the conventional Jacobian operator defined as J (A, B) =

∂x A∂y B − ∂y A∂x B. The streamfunction at the intermediatelevel p2 is computed as average between the streamfunctions

of the levels p1 and p3, so that the material derivative at thelevel p2 can be expressed as D2

t = 1/2(D1t + D3

t ).We choose the following simple functional form for the

diabatic heating:

Q2 = νN C p(T ? − T

)= νN C p

f0 p2

R

(2τ ?

δp−ψ1 − ψ3

δp

), (4)

where the temperature T is evaluated at the pressure level 2 andis defined via hydrostatic relation, which implies that the systemis relaxed towards a prescribed temperature profile T ? with acharacteristic time scale of 1/νN . T ? and τ ? are respectivelydefined as follows:

T ? =TE

2cos

(πyL y

), τ ? =

Rf0

TE

4cos

(πyL y

), (5)

so that TE is the forced temperature difference between the lowand the high latitude border of the domain. In our simulationswe assume no time dependence for the forcing parameter TE ,thus discarding the seasonal effects. Since by thermal windrelation (Eu1 − Eu3) /δp = k × E∇/ (ψ1 − ψ3) /δp, we have thatthe diabatic forcing Q2 in (4) causes a relaxation of the verticalgradient of the zonal wind u1−u3 towards the prescribed profile2m?

= 2dτ ?/dy, where the constant 2 has been introduced forlater convenience.

By imposing ω0 = 0 (top of the atmosphere) andassuming ω4 = −E0∆Hψ3 (Ekman pumping [58]), and aftera rearrangement of Eqs. (1)–(3), one obtains the evolutionequations for the baroclinic field τ = 1/2 (ψ1 − ψ3) and thebarotropic field φ = 1/2 (ψ1 + ψ3):

∂t∆H τ −2

H22∂tτ + J

(τ,∆Hφ + βy +

2H2

2φ

)+ J (φ,∆H τ)

=2νE

H22

∆H (φ − τ)−2κH2

2∆H τ +

2νN

H22

(τ − τ ?

), (6)

∂t∆Hφ + J (φ,∆Hφ + βy)+ J (τ,∆H τ)

= −2νE

H22

∆H (φ − τ) (7)


Table 1Variables of the system and non-dimensionalization factors

Variable Scaling factor Value of scaling factor

x l 106 my l 106 mt u−1l 105 sψ1 ul 107 m2 s−1

ψ3 ul 107 m2 s−1

φ ul 107 m2 s−1

τ ul 107 m2 s−1

A1n ul 107 m2 s−1

B1n ul 107 m2 s−1

A2n ul 107 m2 s−1

B2n ul 107 m2 s−1

m u 10 ms−1

U u 10 ms−1

mn u 10 ms−1

Un u 10 ms−1

wn l−1 10−6 m−1

Lag u−1l 105 sλ j ul−1 10−5 s−1

tp u−1l 105 sT ul f0 R−1 3.5 KE u2l2(δp)g−1 5.1 × 1017 J

For A1n , B1

n , A2n , B2

n , mn , Un , and wn , the index n ranges from 1 to JT . For λ j ,n = 1, . . . , 6 × J T .

where νE = f0 E0 H22 / (2δ) is the viscous-like coupling

between the free atmosphere and the planetary boundary layervia Ekman pumping, and the meaning of τ ? is made clear.Notice that this system features only quadratic nonlinearities.

The two-layer QG system (6) and (7) can be brought to thenon-dimensional form, which is more usual in the meteorolog-ical literature and is easily implementable in computer codes.This is achieved by introducing length and velocity scales l andu and performing a non-dimensionalization of both the systemvariables (x, y, t, φ, τ, T ) (as described in Table 1) and of thesystem constants (Table 2); in our case appropriate values arel = 10 m6 and u = 10 ms−1.

In this work we consider a simplified spectral version of Eqs.(6) and (7), where truncation is performed in the zonal Fouriercomponents so that only the zonally symmetric componentand one of the non-symmetric components are retained. The

derivation is reported in [54,78]. The main reason for thischoice is that we wish to focus on the interaction betweenthe zonal wind and waves, thus neglecting the wave–wavenonlinear interactions. Since quadratic nonlinearities generateterms with Fourier components corresponding to the sum anddifference of the Fourier components of the two factors, wecan exclude direct wave–wave interactions provided that weonly retain a single wave component (see, e.g., [52] for ageneral discussion of these effects in a different context). Notethat if cubic nonlinearities were present, direct self-wave–waveinteraction would have been possible [5]. In the present case,the wave can self-interact only indirectly through the changesin the values of the zonally symmetric fields. This amountsto building up equations which are almost-linear, in the sensethat the wave dynamics is linear with respect to the zonallysymmetric parts of the fields (i.e., the winds). The wavelengthof the only retained wave component is Lx/6, since we intendto represent the baroclinic conversion processes, which in thereal atmosphere take place on scales of Lx/6 or smaller [21]. Inso doing we are retaining only one of the classical ingredientsof GAC, i.e. the zonal wind–wave interaction.

Both φ and τ are thus determined by three real fields: thezonally symmetric parts and the real and imaginary part ofthe only retained zonal Fourier component. A pseudospectraldecomposition with JT modes is then applied to the resulting6 real fields in the y-direction, yielding a set of 6 × JTordinary differential equations in the spectral coefficients. Forthe truncation order JT we have used the values JT =

8, 16, 32, 64.This is the prototypal model we use as laboratory for

the analysis of the GAC. The only form of realism we aretrying to achieve here is that the statistical properties of thenonlinear cycle baroclinic instability — barotropic & baroclinicstabilization are represented.

3. Dynamical and statistical characterization of the modelattractor

The purpose of this section is to show that the zonalwind–wave system captures the nonlinear process of baroclinicconversion and barotropic–baroclinic stabilization displayingsimple (self-scaling, in fact), smooth, robust global behavior

Table 2Values of the parameters used in this work and adimensionalization factors

Parameter Dimensional value Non-dimensional value Scaling factor Value of scaling factor

Lx 2.9 × 107 m 29 l 106 mL y 107 m 10 l 106 mχ 2π/(4.833 × 106) m−1 1.3 l−1 10−6 m−1

H2 7.07 × 105 m 7.07 × 10−1 l 106 mf0 10−4 s−1 10 ul−1 10−5 s−1

β 1.6 × 10−11 m−1 s−1 1.6 ul−2 10−11 m−1 s−1

νE 5.5 × 105 m2 s−1 5.5 × 10−2 ul 107 m2 s−1

κ 2.8 × 105 m2 s−1 2.8 × 10−2 ul 107 m2 s−1

νN 1.1 × 10−6 s−1 1.1 × 10−1 ul−1 10−5 s−1

TE 28–385 K 8–110 ul f0 R−1 3.5 K

Our system is equivalent to that of Malguzzi and Speranza [78], where the following correspondences hold 1/(H22 ) ↔ F , (2νE )/(H2

2 ) ↔ νE/2, (2κ)/(H22 ) → νS ,

and νN ↔ νH .


Table 3Approximate values of the parameter TE where the Hadley equilibrium losesstability via Hopf bifurcation (T H

E ) and where the onset of the chaotic regimeoccurs (T crit

E ) for each of the considered orders of truncation JT

JT T HE T crit

E

8 7.83 9.14816 8.08 8.41532 8.28 8.52264 8.51 8.663

See text for details.

for parameter values of physical relevance. In Section 3.1 webriefly sketch the routes to chaos in our model. The scalinglaws in the fully chaotic regime are then analyzed by studyingthe Lyapunov exponents (Section 3.2) and the bounding box(Section 3.3).

3.1. Bifurcations at the transition to chaos

The system of equations (6) and (7) has the followingstationary solution for zonally symmetric flows:

φ (y) = τ (y) , (8)

2κH2

2

d2τ (y)dy2 +

2νN

H22

(τ (y)− τ ? (y)

)= 0. (9)

Considering the functional form (5) for τ ? (y), the followingtemperature profile T (y) is realized:

T (y) =TE

2

cos(πyL y

)1 +

κνN

(πL y

)2 =T ? (y)

1 +κνN

(πL y

)2 . (10)

This solution describes a zonally symmetric circulationcharacterized by the stationary balance between the horizontaltemperature gradient and the vertical wind shear, whichcorresponds on the Earth system to the idealized pattern of theHadley equilibrium [36,58].

There is a value of the equator-to-pole temperature gradientT H

E such that if TE < T HE the Hadley equilibrium (8)

and (9) is stable and has an infinite basin of attraction,whereas if TE > T H

E it is unstable. In the stableregime with TE < T H

E , after the decay of transients, thefields φ, τ , and T are time independent and feature zonalsymmetry — they only depend on the variable y. Moreover,they are proportional by the same near-to-unity factorto the corresponding relaxation profiles, compare (8)–(10).In particular, this implies that all the equilibrium fields areproportional to the parameter TE .

When increasing the values of the control parameter TEbeyond T H

E , the equilibrium described by (8) and (9) becomesunstable: a complex conjugate pair of eigenvalues of thelinearization of (A.4)–(A.9) at the equilibrium (8) and (9)cross the imaginary axis and their real part turns positive.This suggests the occurrence of a Hopf bifurcation. Thecorresponding physical scenario is the following: for highvalues of the meridional temperature gradient the Hadleyequilibrium is unstable with respect to the process of baroclinic

conversion, which allows the transfer of available potentialenergy of the zonal flow stored into the meridional temperaturegradient into energy of the eddies [12,19].

A stable periodic orbit branches off from the Hadleyequilibrium (8) and (9) as TE increases above T H

E . Theattracting periodic orbit persists for a narrow interval of TE ,and for slightly larger values of TE , i.e. TE > T crit

E , astrange attractor appears. Describing in detail the route tothe formation of the strange attractor is beyond the scopeof the present paper: this problem will be tackled elsewhere.Suffice here to say that, in the case of JT = 16, 32 and64, this route involves the breakdown of a two-torus attractorof the flow and is very similar to those described in [9,61].The observed values of T H

E and T critE typically increase with

the considered truncation order JT . Results are reported inTable 3 for the choice of constants reported in Table 2. A finerresolution allows for more efficient stabilizing mechanisms,which counteract the baroclinic instability, because they actpreferentially on the small scales. Such mechanisms are thebarotropic stabilization of the jet, increasing the horizontalshear through the convergence of zonal momentum, whichis proportional to the quadrature of the spatial derivativesof the fields φ and τ , and the viscous dissipation, which isproportional to the laplacian of the fields φ and τ . This is aclarifying example that in principle it is necessary to includesuitable renormalizations in the parameters of a model whenchanging the resolution JT , in order to keep correspondencewith the resulting dynamics [50]. In our case the values of T H

Eand T crit

E obtained for the adopted resolutions are neverthelessrather similar. In a related framework, see [29] for a detaileddiscussion of the effects of changing the truncation order forthe structure of the bifurcations of a model.

We suspect that plenty of unstable periodic orbits andinvariant tori coexist with the attractor of model (A.4)–(A.9)for sufficiently large TE . Indeed, from the right panel ofFig. 2 we deduce that the Hadley equilibrium undergoesseveral other bifurcations after the first one. Since thenumber of unstable eigenvalues of the Hadley equilibriumincreases at each Hopf bifurcation, the periodic orbits thatbranch off have unstable manifolds of increasingly highdimension. Moreover, these unstable periodic orbits in turnundergo Hopf bifurcations, where unstable two-tori branch off,compare [78, Sec. 5]. It seems, therefore, that the phase spacequickly gets crowded with high dimensional unstable invariantmanifolds.

3.2. Lyapunov exponents and dimension of the strangeattractor

To characterize the dynamical properties of the strangeattractors of (A.4)–(A.9) we resort to the study of the Lyapunovexponents [20,57], denoted by λ1 ≥ λ2 ≥ · · · ≥ λN , N =

6 × JT . The maximal exponent λ1 becomes positive as TEcrosses the torus breakdown value T crit

E , and then increasesmonotonically with TE .

The spectrum of the Lyapunov exponents is plotted in theleft panel of Fig. 3 for three different values of TE , again with


Fig. 3. Left: Spectrum of the Lyapunov exponents for TE = 9, TE = 30, and TE = 110. Units as for λ j and TE as described in Table 1. Right: Lyapunov dimensionof the attractor of (A.4)–(A.9) as a function of TE for JT = 8, 16, 32, and 64. All the straight lines are parallel and the domain of validity of the linear fit is apparentlyhomothetic.

JT = 32. Note that despite the great simplifications adoptedin this work, the obtained Lyapunov spectra are qualitativelysimilar to what reported in [75,80], where a much larger numberof degrees of freedom was considered. The distribution of theexponents approaches a smooth shape for large TE and a similarshape is observed for JT = 64 (not shown). This suggests theexistence of a well-defined infinite baroclinicity model obtainedfrom (6) and (7) as a (possibly, singular perturbation) limitfor TE → ∞. We will analyze elsewhere this mathematicalproperty.

3.2.1. Dimension of the strange attractorThe Lyapunov exponents are used to compute the Lyapunov

dimension (also called Kaplan–Yorke dimension, see [20,41]) and metric entropy (also known as Kolmogorov–Sinaientropy [20]). The Lyapunov dimension is defined by

DL = k +

k∑j=1

λ j

|λk+1|, (11)

where k is the unique index such that∑k

j=1 λ j ≥ 0 and∑k+1j=1 λ j < 0. Under general assumptions on the dynamical

system under examination, DL is an upper bound for theHausdorff dimension of an attractor.

We have also computed (not shown) other numericalestimates for the dimension of an attractor: the correlationand information dimensions [25]. However, these estimatesbecome completely meaningless when the Lyapunov dimensionincreases beyond, say, 20, and issues of reliability arisealso below 10. In particular, the correlation and informationalgorithms drastically underestimate the dimension. This is awell-known problem: for large dimensions, prohibitively longtime series have to be used. Ruelle [66] suggests as a ruleof thumb that a time series of 10d statistically independentdata are need in order to estimate an attractor of dimension d.Therefore, computational time and memory constraints in factlimit the applicability of correlation-like algorithms only to lowdimensional attractors.

Both the number of positive Lyapunov exponents and theLyapunov dimension increase with TE . As shown in the right

panel of Fig. 3, for all the considered values of JT , it is possibleto distinguish three regimes of variation of DL as a function ofTE :

• For small values of (TE −T critE ), DL ∝ (TE −T crit

E )γ , with γranging from ∼0.5 (JT = 8) to ∼0.7 (JT = 64). The rangeof TE where this behavior can be detected increases with JT .

• For larger values of TE a linear scaling regime DL ∼

βTE + const. is found. For all JT , the linear coefficient isremarkably close to β ∼ 1.2. The domain of validity of thelinear approximation is apparently homothetic.

• For TE larger than a JT-dependent threshold, there occursa sort of phase–space saturation as the Lyapunov dimensionbegins to increase sublinearly with TE . Note that while forJT = 8 the model is in this regime in most of the exploredTE -domain (TE & 20), for JT = 64 the threshold is reachedonly for TE & 108. Further discussions on this point will begiven in Sections 3.3 and 4.

3.2.2. Metric entropyThe metric entropy h(ρ) of an ergodic invariant measure ρ

expresses the mean rate of information creation, see [20] fordefinition and other properties. If a dynamical system possessesa SRB invariant measure ρ, then Pesin’s identity holds:

h(ρ) =

∑λ j>0

λ j . (12)

However, existence of an SRB measure is rather difficult todemonstrate for a given non-hyperbolic attractor [20,64]. Atpractical level it is enough to invoke the beneficial role of asmall numerical noise [65]; we then refer to the sum of thepositive Lyapunov exponents as metric entropy.

The maximal Lyapunov exponent and the metric entropy asfunctions of TE are compared for JT = 8, 16, 32, and 64 inFig. 4. It turns out that, for fixed JT , λ1 increases sublinearlywith TE , whereas the metric entropy has a marked lineardependence h ∼ β(TE − T crit

E ), with β ranging from ∼0.15(JT = 8) to ∼0.5 (JT = 64). Moreover, for a given value ofTE , the metric entropy increases with JT , whereas for TE fixed,λ1 decreases for increasing values of JT . From the dynamicalviewpoint, this means on one hand that the maximal sensitivity


Fig. 4. Left: Maximal Lyapunov exponent on the attractor of (A.4)–(A.9) as a function of TE for JT = 8, 16, 32, 64. Right: Metric entropy. Linear dependencesh ∼ β(TE − T crit

E ) occur for all values of JT .

of the system to variations in the initial condition along a singledirection is largest for JT = 8. On the other hand, there aremany more active degrees of freedom for JT = 64 and theycollectively produce a faster forgetting of the initial conditionas time goes on.

A more precise assessment of the time scales of the systemand of predictability fluctuations could be gained by computinggeneralized Lyapunov exponents and performing the relatedmultifractal diagnostics [4,7,18], which is outside the scope ofthis paper.

3.3. Bounding box of the attractor

The bounding box of a set of points in an N -dimensionalspace is defined as the smallest hyperparallelepiped containingthe considered set [74]. For clarity, in the N -dimensional phasespace, where N = 6 × JT , the volume VB B is computed as:

VB B =

N=6×JT∏k=1

[max

ttr<t<tmax(zk (t))− min

ttr<t<tmax(zk (t))

]. (13)

Here the zk denote the 6 × JT variables spanning the phasespace of the system, in our case the Fourier coefficients A1

j , A2j ,

B1j , B2

j , m j , and U j , with j = 1, . . . , JT . The condition t > ttrallows for the transients to die out. Typically, ttr is rather safelyfixed to 1500, which correspond to about five years.

When the Hadley equilibrium is the universal attractor, thevolume VB B is zero, while it is non-zero if the computed orbitis attracted to a periodic orbit, a two-torus or a strange attractor.We observe that the volume of the bounding box is not anindicator of chaoticity, but only provides a measure of the bulksize of the attractor in phase space. In our case, it turns outthat VB B always grows very regularly with TE . Specifically,each of the factors in the product (13) increases with TE , sothat expansion occurs in all directions of the phase space. Thismatches the basic expectations on the behavior of a dissipativesystem having a larger input of energy.

In the right panel of Fig. 5 we present a plot of log(VB B) asfunction of TE for the selected values of JT = 8, 16, 32, and 64.In the case JT = 8, VB B obeys with great precision the power

Table 4Power-law fits of the volume of the bounding box as VB B ∝ (TE − T crit

E )γ intwo different ranges of TE −T crit

E for each of the considered orders of truncationJT

JT γ [log(TE − T critE ) ≤ 0.5] γ [log(TE − T crit

E ) ≥ 0.5]

8 40±1 40±116 33±3 80±132 66±2 160±164 133±4 320±1

See text and Fig. 5 for details.

law VB B ∝ (TE − T critE )γ in the whole domain TE ≥ 9. The

best estimate for the exponent is γ ∼ 40. Given that the totalnumber of Fourier components is 6 × JT = 48, this impliesthat the growth of the each side of the bounding box is on theaverage proportional to about the 5/6th power of (TE − T crit

E ).For higher values of JT , two sharply distinct and well-

defined power-law regimes occur. For JT = 16, in the lowerrange of (TE − T crit

E ) – corresponding in all cases to TE .T crit

E + 1.5 – the volume of the bounding box increases withabout the 35th power of (TE − T crit

E ), while in the upper rangeof (TE − T crit

E ) – for TE & T critE + 1.5 – the power-law

exponent abruptly jumps up to about 80. For JT = 32 thesame regimes can be recognized, but the values of the bestestimates of the exponents are twice as large as what obtainedwith JT = 16. Similarly, for JT = 64 the best estimates of theexponents are twice as large as for JT = 32. The results onthe power-law fits of VB B ∝ (TE − T crit

E )γ are summarizedin Table 4. We emphasize that in all cases the uncertaintieson γ , which have been evaluated with a standard bootstraptechnique, are rather low and total to less than 3% of the bestestimate of γ . Moreover, the uncertainty of the power-law fitgreatly worsens if we detune the value of T crit

E by as little as0.3, thus reinforcing the idea that fitting a power law againstthe logarithm of (TE − T crit

E ) is a robust choice.When considering separately the various sides of the

bounding box hyperparallelepiped (not shown), i.e., each ofthe factors in the product (13), we have that for JT = 8 allof them increase as about (TE − T crit

E )5/6 in the whole range.


Fig. 5. Left: Volume of the bounding box VB B of the attractor as a function of the detuning parameter TE − T critE for JT = 8, 16, 32, 64. For description of the

power-law fits, see the text and Table 4. Right: Value of the corresponding sides of the bounding box pertaining to the variables A1j for JT = 32 (red lines) and 64

(magenta lines). Notice the two power-law regimes mentioned in the text. (For interpretation of the references to colour in this figure legend, the reader is referredto the web version of this article.)

For JT = 16, 32, and 64, in the lower range of TE eachside of the bounding box increases as about the 1/3rd powerof (TE − T crit

E ), while in the upper range of TE each sideof the bounding box increases as about the 5/6th power of(TE − T crit

E ). Selected cases are depicted in the right panelof Fig. 5. So for a given value of truncation order JT , theratios between the ranges of the various degrees of freedomare essentially unchanged when varying TE , so that the systemobeys a sort of self-similar scaling with TE .

Summarizing, for sufficiently high truncation order (JT ≥

16) a robust parametric dependence is detected for the volumeof the bounding box as a function of TE :

VB B ∝ (TE − T critE )γ , γ = εN

ε ∼

{1/3, TE − T crit

E . 1.5,5/6, TE − T crit

E & 1.5,(14)

where N = 6 × JT is the number degrees of freedom.The comparison, for, say, JT = 16 and 32, of factors

in (13) having the same order for the same value of TE − T critE

provides insight about the sensitivity to model resolution. Inthe following discussion, we examine the variables A1

j butsimilar observations apply to all other variables A2

j , B1j , B2

j ,U j , and m j . The factors related to the gravest modes, suchas [max(A1

j (t)) − min(A11 (t))], agree with high precision,

thus suggesting that the large scale behavior of the systemis only slightly affected by variation of model resolution.When considering the terms related to the fastest latitudinallyvarying modes allowed by both truncation orders, such as[max(A1

j (t)) − min(A1j (t))] with 17 ≤ j ≤ 32, we have that

those obtained for JT = 32 are larger than the correspondingfactors obtained for JT = 64, and the distance between pairsof the same order increases with j . See the right panel ofFig. 5. This is likely to be the effect of spectral saturation:the dynamics contained in the scales which are resolved in thehigher resolution models are projected in the fastest modes ofthe model with lower resolution. The same effect is observed

when comparing, for JT = 16 and 32, coefficients of the sameorder such as [max(A1

j (t)) − min(A1j (t))] with 9 ≤ j ≤ 16.

The JT = 8 case does not precisely match this picture.

4. Statistical properties of the total energy and zonal wind

In terms of studying the model (A.4)–(A.9), the analysisof the statistics of observables having a direct physicalsignificance is complementary to the investigation performedin Section 3.2, where we have considered properties of specialrelevance for a dynamical system-oriented analysis. Today,concepts and tools borrowed from dynamical systems theoryhave widespread applications in the context of modern weatherforecast methods, where specific strategies are consideredfor the local exploration of the phase space of atmosphericmodels [40]. Since we want to deal with easily readableglobal physical quantities, we consider the total energy and theaverage zonal wind of the model.

4.1. Total energy

The total energy of the system E(t) is a global observableof obvious physical significance, statistically obeying a balancebetween the external forcing and the internal and surfacedissipation. The horizontal energy density of the two-layer QGsystem can be expressed as follows:

e (x, y, t) =δpg

[12

(E∇ψ1

)2

+12

(E∇ψ3

)2+

12H2

2(ψ1 − ψ3)

2

]

=δpg

[(E∇φ)2

+

(E∇τ)2

+2

H22τ 2

]. (15)

Here the factor δp/g is the mass per unit surface in each layer,the last term and the first two terms inside the brackets represent


Fig. 6. Left: E(t) for the Hadley equilibrium (black line) and deduced from the observed fields in the chaotic regime for JT = 64 (magenta line); the magentadashed line delimit the σ -confidence interval. Right: fractional deviations of E(t), for JT = 8, 16, and 32, with respect to JT = 64. See text for details. (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the potential and kinetic energy, respectively, thus featuring aclear similarity with the functional form of the energy of aharmonic oscillator. Note that in (15) the potential energy termis half of that reported in [58], which contains a trivial algebraicmistake in the derivation of the energy density, as discussedwith the author of the book. We then consider as observable thetotal energy E(t), evaluated by integrating the energy densityexpression (15):

E(t) =

∫ L y

0

∫ Lx

0e(x, y, t)dxdy

= 6∫ L y

0

∫ 2πχ

0e(x, y, t)dxdy. (16)

Potential energy is injected into the system by zonallysymmetric baroclinic forcing τ ?, part of it is transformedinto wave kinetic energy by baroclinic conversion, and kineticenergy is eventually dissipated by friction such as thatdetermined by Ekman pumping. This constitutes the Lorenzenergy cycle [45,59], which has been analyzed for this systemin [78]. In Table 1 we report the conversion factor of the totalenergy between the non-dimensional and dimensional units.

For the Hadley equilibrium, the time-independent expres-sion for the total energy is:

E(t) = E =δpg

Lx L y

RTE

4 f0

1

1 +κνE

(πL y

)2

2

×

(π2

L2y

+1

H22

). (17)

The total energy is proportional to T 2E and is mostly (95%)

stored as potential energy, which is described by the secondterm of the sum in (17).

In Fig. 6 we present the results obtained for the variousvalues of JT used in this work. In the left panel we presentthe JT = 64 case, which is representative of that obtained

also in the other cases. The time-averaged total energy ismonotonically increasing with TE , but when the system entersthe chaotic regime, E(t) is much lower than the value for thecoexisting Hadley equilibrium. This behavior may be relatedto the much larger dissipation fuelled by the chaos-drivenactivation of the smaller scales. In the chaotic regime E(t) ischaracterized by temporal variability, which becomes more andmore pronounced for larger values of TE .

In the right panel of Fig. 6 we compare the cases JT = 8,16, 32 with respect to JT = 64. The overall agreement of E(t),where X denotes the time average of the field X , is good butprogressively worsens when decreasing JT: for JT = 32, themaximal fractional difference is less than 0.01, while for JT =

8 it is about one order of magnitude larger. Differences amongthe representations given by the various truncations levels alsoemerge in power-law fits such as E(t) ∝ T γE . In the regimewhere the Hadley equilibrium is attracting, this fit is exact, withexponent γ = 2. For TE − T crit

E . 1.5 and TE > T HE (the value

of the first Hopf bifurcation, see Table 3), for all the values of JTthe power-law fit is good, with γ = 1.90±0.03, so that a weaklysubquadratic growth is realized. For TE − T crit

E & 1.5, onlythe JT = 32 and 64 simulations of E(t) obey with excellentapproximation a weaker power law, with γ = 1.52 ± 0.02 inboth cases, while the cases JT = 8 and 16 do not satisfactorilyfit any power law.

The agreement worsens in the upper range of TE , whichpoints at the criticality of the truncation level when strongforcings are imposed. Nevertheless, the observed differencesare strikingly small between the cases, say, JT = 8 andJT = 64, with respect to what could be guessed by lookingat the Lyapunov dimension, metric entropy, and bounding boxvolume diagnostics analyzed in the previous sections.

4.2. Zonal wind

Let U (y, t) = 1/2 (u1(y, t)+ u3(y, t)) be the zonal averageof the mean of the zonal wind at the two pressure levels p1


Fig. 7. Left: 〈U 〉 = 〈m〉 for the Hadley equilibrium (black line) and deduced from the observed fields in the chaotic regime for JT = 64 (magenta line); the magentadashed and dotted lines delimit the σ -confidence interval for 〈U 〉 and 〈m〉, respectively. Right: fractional deviations of 〈U 〉 = 〈m〉 for JT = 8, 16, and 32 withrespect to JT = 64. See text for details. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

and p3 and m(y, t) = 1/2 (u1(y, t)− u3(y, t)) be the zonalaverage of the halved difference of the zonal wind at the twopressure levels p1 and p3. We here examine the latitudinalaverage, denoted by 〈•〉, of U and m:

〈U (y, t)〉 =1L

∫ L

0U (y, t)dy =

2π

JT∑j=1, j odd

U j

j, (18)

〈m(y, t)〉 =1L

∫ L

0m(y, t)dy =

2π

JT∑j=1, j odd

m j

j. (19)

We have that 〈U (y, t)〉 is proportional to the total zonal momen-tum of the atmosphere, whereas 〈m(y, t)〉, by geostrophy, isproportional to the zonally averaged temperature difference be-tween the northern and the southern boundaries of the domain.Computation of such space averages at the time-independentHadley equilibrium is straightforward:

〈m(y)〉 = 〈U (y)〉 =R

f0L y

TE

21

1 +κνN

(πL y

)2 . (20)

Since we cannot have net, long-term zonal forces acting on theatmosphere at the surface interface, the spatial average of thezonal wind at the pressure level p3 must be zero. Therefore, theoutputs of the numerical integrations must satisfy the constraint〈m(y, t)〉 = 〈U (y, t)〉, which is automatically satisfied at theHadley equilibrium. The results of the integration – wheresuch constraint is obeyed within numerical precision – arepresented in Fig. 7. In the left panel we plot the outputsfor JT = 64, which, similarly to the total energy case, iswell representative of all the JT cases. The average winds aremonotonically increasing with TE , but, when the system entersthe chaotic regimes, the averages 〈m(y, t)〉 = 〈U (y, t)〉 havea much smaller value than at the Hadley equilibrium, and theydisplay sublinear growth with TE . Moreover, for TE > T crit

E thetemporal variability of the time series 〈m(y, t)〉 and 〈U (y, t)〉increases with TE . The variability of 〈m(y, t)〉 results to be

slightly larger than that of 〈U (y, t)〉, probably because the latteris related to a bulk mechanical property of the system such asthe total zonal momentum.

The effects of lowering JT are illustrated in Fig. 7 right.The overall agreement, expressed by a small value of thefractional differences, progressively worsens for smaller JT .Notice the similarity of the functional shapes with Fig. 6 right.The results in Fig. 7 right can be summarized as follows:the coarser-resolution models have higher total temperaturedifference between the two boundaries for values of TE up toabout 30 and lower temperature differences for higher valuesof TE . This implies that while for TE . 30 the latitudinal heattransport increases with JT as a positive trade-off between thehigher number of unstable baroclinic modes (within a sloppylinear thinking) or, better, smaller scale baroclinic conversionprocesses taking place in a higher dimensional attractor, and theenhancement of the barotropic and viscous stabilizing effects,for TE & 30 the converse is true.

Again, differences between the various truncations levelsemerge as one attempts power-law fits of the form 〈m(y, t)〉 =

〈U (y, t)〉 ∝ T γE . For the Hadley equilibrium regime we haveγ = 1. For TE . 10 and above the first Hopf bifurcation, for allvalues of JT the power-law fit is good, with γ = 0.875±0.005.For TE − T crit

E & 1.5, only the simulations with JT = 32 and64 obey a power law (with γ = 0.58 ± 0.02) with excellentapproximation, while the realizations of the JT = 8 and 16cases do not fit any power law.

Since we are dealing with a QG system, these observationson the wind fields imply that while the time-averagedmeridional temperature difference between the northern andsouthern boundary of the system increases monotonicallywith TE , as to be expected, the realized value is greatlyreduced by the onset of the chaotic regime with respect tothe corresponding Hadley equilibrium. This is the signatureof the negative feedback due to the following mechanism:when the poleward eddy transport of heat is realized, itcauses the reduction of the meridional temperature gradient,thus limiting by geostrophy the wind shear, which causes in


Fig. 8. Time-averaged latitudinal profiles U (y) (solid lines) and m(y) (dashed lines). In all figures the black solid line indicates the U (y) = m(y) profile of theHadley equilibrium, the blue and red lines refer to the cases JT = 8 and JT = 32, respectively. The values of TE are indicated. Note that the vertical scale forTE = 9 and 10 is about 1/2 as for the other two figures. (For interpretation of the references to colour in this figure legend, the reader is referred to the web versionof this article.)

turn a reduction of baroclinically induced eddies towards themarginal stability [79]. This process can be considered to bethe nonlinear generalization of the baroclinic adjustment [79],which implies that the system statistically balances near anaverage state which corresponds to a fixed point which isneutral with respect to baroclinic instability. This type ofadjustment often relies on the (sometimes hidden) idea that avariational principle holds concerning the relationship betweenthe heat & momentum fluxes and the basic state gradientas, for example, in classical convection for which it can beproved that the most unstable mode is the one carrying heatmost efficiently across the basic state gradient. In fact, sucha variational principle does not hold for ordinary baroclinicinstability, as it can be proved that the most rapidly growingbaroclinic mode is not the one with the highest heat flux. Butin this model, as opposed to the general case, the variationalassumption in question is essentially correct, since only onezonal wave is considered, and so the fastest growing unstablewave is also the wave transporting northward the largest amountof heat [76,78]. Nevertheless, the adjustment mechanism doesnot keep the system close to marginal stability since for TE >

T critE both the instantaneous and the time-averaged fields of

the system are completely different from those realized at the

Hadley equilibrium. Also, this result denies the possibility thatthe time-mean circulation is maintained by eddies which can beparameterized in terms of the time-mean fields.

By examining more detailed diagnostics on the winds,such as the time-averaged latitudinal profiles of U (y) andof m(y) (Fig. 8), relevant differences are observed betweenJT = 8 and the other three cases. Results are presented forJT = 8 and JT = 32, the latter being representative alsoof JT = 16 and 64. We first note that already for TE = 9and 10, such that only a weakly chaotic motion is realized,the U (y) and m(y) profiles feature in both resolutions relevantqualitative differences with respect to the corresponding Hadleyequilibrium profile, although symmetry with respect to thecenter of the channel is obeyed. The U (y) and m(y) profilesare different (the constraint 〈m(y, t)〉 = 〈U (y, t)〉 being stillsatisfied), with U (y) > m(y) at the center and U (y) < m(y) atthe boundaries of the channel. Nevertheless, like for the Hadleyequilibrium, both U (y) and m(y) are positive and are larger atthe center of the channel than at the boundaries. Consequently,at pressure level p1 there is a westerly flow at the center of thechannel and easterly flows at the two boundaries, and that atpressure level p3 the wind is everywhere westerly and peaks atthe center of the channel. Such features are more pronounced


for the JT = 32 case, where the mechanism of the convergenceof zonal momentum is more accurately represented.

For larger values of TE , the differences between the twotruncation levels become more apparent. For JT = 8, theobserved U (y) and m(y) profiles tend to flatten in the centerof the channel and to become more similar to each other.Therefore, somewhat similarly to the Hadley equilibrium case,the winds at the pressure level p1 tend to vanish and allthe dynamics is restricted to the pressure level p3. The m(y)profiles for JT = 32 are quite similar to those of JT = 8,even if they peak and reach higher values in the center of thechannel and are somewhat smaller at the boundaries. So whena finer resolution is used, a stronger temperature gradient isrealized in the channel center. The U (y) profiles obtained forJT = 32 are instead very different. They feature a strong,well-defined peak in the channel center and negative valuesnear the boundaries. Therefore, the winds in the upper pressurelevel are strong westerlies, and peak in the center of thechannel, while the winds in the lower pressure level feature arelatively strong westerly jet in the center of the channel andtwo compensating easterly jets at the boundaries. The fact thatfor higher resolution the wind profiles are less smooth and havemore evident jet-like features is related to the more efficientprocess of eddy zonal momentum flux convergence [43,72],which keeps the jet together, and is related to barotropicstabilization [56]. Examination of the latitudinal profiles inFig. 8 clarifies our choice to extend the latitudinal domain of themodel beyond the geometrically and geographically realisticmid-latitude channel. Due to this, the wind fields in the centralportion of the domain (the latter corresponds to mid-latitudesand is of primary interest in this work), are rather different thanat the boundary regions.

5. Parametric smoothness and self-scaling of the attractorproperties with respect to TE: Modeling the atmospheric jet

If general enough, the scaling properties discussed inSections 3 and 4 could be of great help in setting up a theory forthe overall statistical properties of the GAC and in guiding – ona heuristic basis – both data analysis and realistic simulations. Aleading example for this would be the possibility of estimatingthe sensitivity of the model output with respect to changes inthe parameters. Physical insight into the relevant mechanismscan be obtained by considering the main feedback mechanismssetting the average statistical properties of the system, e.g. whenchanging the forcing TE from TE = T 0

E > T critE to TE =

T 0E + ∆TE . The change in TE directly causes an increase in

the value of m1 through Newtonian forcing and indirectly anincrease in the value of U1 through equilibration driven byEkman pumping. The increase in the value of m1 enhances thebaroclinic conversion and so the values of A1

j , A2j , B1

j , and B2j .

The eddy stresses, due to the quadratures of the A and B fields,act as feedback by decreasing the value of the m fields throughdepletion of baroclinicity and by increasing the value of Ufields through barotropic convergence, so that the feedback loopis closed. In the chaotic regime, these feedback mechanismscan be thought as constituting the above mentioned generalized

type of statistical baroclinic adjustment process, which reducesthe average value of the equator-to-pole temperature difference.This process is not reducible to the stationary balances of theclassic theory of GAC, since it takes place when the systemlives in a strange attractor and for sure adheres more closelyat conceptual level to the equilibration of the atmosphere,given the chaotic nature of the latter. The power laws obtainedare the macroscopic realization of the envisioned statisticalbaroclinic adjustment process, which determines the nature andthe statistical properties of the system attractor. Preliminarycalculations performed by the authors together with Vannitsem(unpublished results) suggest that the power laws presentedhere hold in general also on simplified yet global models of theatmospheric circulation, so the obtained scalings laws might bevery helpful in establishing a sort of bulk climate theory for theatmospheric disturbances. Apart from the diagnostic indicatorsexamined in the previous sections, other statistical propertiessuch as, notably, the distribution of extremes of some fieldvariables of the system display similar properties of smoothnessand limited variation, as well as being expressed as power-lawscalings of TE [26]. The smoothness property proves to be ofespecially great help when dealing with statistical inference inthe presence of imposed trends in a non-autonomous version ofthe dynamical system here considered [27].

The fact that all the considered dynamical indicators(Lyapunov exponents and dimension, volume of the attractorbounding box) and physical quantities (total energy and meanzonal wind) feature a smooth, simple dependence with respectto the forcing parameter TE and a limited variation is at firststriking if one keeps in mind the phenomena typically occurringin systems having low dimensional non-hyperbolic strangeattractors. Examples are provided by equations modelinglaser systems: parameter regions where chaotic output occursare very often interspersed with windows of periodicity, seee.g. [32]. In many such cases, the system properties arediscontinuous at the boundaries of the phase-locking intervals,since the attracting orbit vanishes (typically through a saddle-node bifurcation) and a strange attractor appears at once.Other examples of discontinuous dependence on externalparameters are contained in the enormous literature on attractorcrises (see [63] and references therein). Analogous types ofbifurcations occur in many systems having low dimensionalstrange attractors: the list includes Henon-like families [30,73],climate models (see e.g., [9] and references therein), infinitedimensional systems [24]. Also compare [20] and referencestherein.

The general, crucial point is here that the effect of a lackof smoothness, or anyway of a very sensitive dependenceof model statistical properties on the external parameters, oron details of the model representation, would be detrimentalfor our ability in setting up a GAC theory, as well asentailing dramatic consequences on our ability to model theatmosphere for practical purposes as well. For the presentmodel (A.4)–(A.9), no window of periodicity or any otherstatistical pathology were detected in the smooth scaling range(say, TE > 16), independently of the truncation order. Wehave also tried slightly different spectral discretization schemes


and integration methods (such as leapfrog or Runge–Kutta4), but this feature persisted in all cases. The behaviorof the system matches the conjectures stated in [1] forgeneral dissipative systems, basically proposing that in spiteof the lack of structural stability, the topological changesinduced in the attractor by parameter variations are notcatastrophic if the dimension of the system is high enough.The implication of the conjectures is that, as the dimensionof a dissipative dynamical system is increased, the numberof positive Lyapunov exponents increases monotonically, theLyapunov exponents tend towards continuous change withrespect to parameter variation and the number of observableperiodic windows decreases. In substance, it would appear thatthe modelers paradigm described in the introduction could havesome rigorous justification. Persistent chaos and smooth-likedependence of the SRB measure with respect to an externalparameter thus appears to be typical of high dimensionalsystems [2]. One of the first remarks hinting at this generalproperty is in [24].

6. Summary, conclusions and future developments

We have described the construction and the dynamicalbehavior of an intermediate complexity model of theatmospheric system. We take as numerical laboratory for theGAC a quasi-geostrophic (QG) model where the mid-latitudeatmosphere is taken as being composed of two layers and theβ-plane approximation is considered.

A single zonal wave solution is assumed and by a spectraldiscretization in the latitudinal direction, the latter equationis reduced to a system of N = 6 × JT ordinary differentialequations, where JT + 1 is the number of nodes of the(latitudinally speaking) fastest varying base function. We haveconsidered the cases JT = 8, 16, 32, and 64. Althoughrelevant ingredients of geometrical and dynamical natureof the real atmospheric circulation are still missing in thissimplified theoretical representation, our model features somefundamental processes determining the general circulation ofthe Earth atmosphere. In particular, the acting processes are thebaroclinic conversion, transforming available potential energyinto waves; the nonlinear stabilization of the zonal jet; andthe thermal diffusion and Ekman pumping-driven viscousdissipation. It is clear that realism in the representation of theatmosphere is not an issue in this work.

When a larger pool of available energy is provided,the dynamics of the system is richer, since the baroclinicconversion process can transfer larger amounts of energy tothe disturbances. Correspondingly, by increasing the parameterTE , which describes the forced equator-to-pole temperaturegradient and parameterizes the baroclinic forcing, the overallbehavior of the system is greatly altered. The attractor changesfrom a fixed point to a strange attractor via a finite numberof bifurcations, starting with a Hopf bifurcation at TE = T H

Edetermining the loss of stability of the Hadley equilibrium, anda final two-torus breakdown at TE = T crit

E . The observed routeto chaos is qualitatively the same for JT = 16, 32, and 64, and

the values T HE and T crit

E weakly depend on JT (Table 3), whilesome differences are observed for JT = 8.

The strange attractor is studied by means of the Lyapunovexponents. The Lyapunov spectra obtained for JT = 32, 64resemble what obtained in more complex QG models [75,80]. A striking feature of this dynamical system is the rathersmooth dependence on the parameter TE of all of its dynamicalproperties. No windows of periodicity have been detected inthe chaotic range and this is quite uncommon especially whencomparing with low dimensional chaotic systems, such as theHenon–Pomeau mapping [37,73] or the Lorenz flow [46]. Themetric entropy, representing the total dynamical instability,increases linearly with TE for TE > T crit

E for all examinedvalues of JT , and is larger for larger values of JT . The Lyapunovdimension DL increases with both TE and JT . In particular, byincreasing TE , initially the dimension grows with a sublinearpower law DL ∝ (TE − T crit

E )γ , followed by a linear scalingregime, while for large TE , DL saturates. The fact that thedimensionality of the attractor increases with the total energyof the system (they are both monotonically increasing withTE ) suggests that the system has a positive temperature, in astatistical mechanical sense. For JT ≥ 16, each side of thebounding box of the attractor increases as ∝ (TE − T crit

E )1/3 forTE −T crit

E . 1.5 and as ∝ (TE −T critE )5/6 for larger values of TE ,

while for JT = 8 only the latter regime is present. Therefore,the ratios of the ranges of the various degrees of freedom remainessentially unchanged when varying TE , yielding a self-similarscaling property.

We emphasize that from the point of view of representing thedynamical properties of the atmosphere, bifurcations are oftenstudied with two goals:

• identifying sharp regime transitions;• obtaining reduced models by projection of the equations on

the center manifold corresponding to the bifurcation.

In the present model, no sudden and sharp variations (crises)of the attractor have been detected sufficiently far from T crit

E ,i.e., in the range of physically relevant parameter values: herethe properties of the attractor vary quite smoothly with theexternal parameter TE . Furthermore, the large dimensionalityof the chaotic attractor leaves little hope that the dynamics canbe reduced by projection on some center manifold. Therefore, itseems very unlikely that the signature of the bifurcations at thetransition to chaos, of the kind exploited e.g. in the context ofphysical oceanography in a recent book [23], may be detectablein the fully developed strange attractor and may be useful forcomputing some of its statistical properties.

Our investigation has focused also on observables of moreimmediate physical interest. When the system enters the chaoticregime, the average total energy and average zonal windshave lower values than those of the corresponding unstableHadley equilibrium, because the occupation of the fastervarying latitudinal modes fuels Ekman pumping-driven viscousdissipation, which acts preferentially on the small scales. Thetotal energy and the average wind field obey with excellentapproximation a subquadratic and sublinear power laws ∝ T γE ,respectively, and are in quantitative agreement for all values


of JT , essentially because these quantities are representative ofglobal balances.

We do not expect to find a global uniform parametricallycontrolled self-scaling of the system statistical properties when,beyond a certain parameter threshold, new physical processescome into play, e.g. in our case allowing for a different anddifferently efficient way of converting available into kineticenergy. The change in the slope observed in all cases forvalue of TE − T crit

E ≈ 1.5 may then be interpreted as achangeover from a quasi-linear baroclinic activity to a fullychaotic regime. A possible source of break-up of self-similarityof the statistical properties of the attractor with respect toexternal parameters is resonant behavior, i.e. the preferentialoccurrence of certain physical processes for a bounded rangeof values of a given parameter. In the case of atmosphericdynamics, this effect might more relevant than the abovementioned onset of windows of periodicity, although in a rangeof variability which is different from what explored in thispaper, namely the low frequency variability [5,6,68]. In thiscontext, using more realistic atmospheric models comprisingbottom orography, a physically interesting experiment to beperformed with the purpose of finding the break-up of self-similarity is to parametrically tune and detune the statisticalonset of the orographic baroclinic energy conversion [10,16,68], which is in nature a rather different process from theconventional baroclinic conversion, acting on much largerscales (order of 10 000 km in the real atmosphere).

We summarize the main conclusions regarding the physicalproperties analyzed in this work as follows:

1. the model displays statistical properties having sufficientlysmooth and slow dependence in the parameter space –no crises – as well as robust with respect to modelrepresentation. The observed persistent chaos and smooth-like dependence of the measure might be related to recentlyestablished conjectures, such as the chaotic hypothesisproposed by Gallavotti and Cohen [17,33] and the ideasproposed in [1,2]. This leaves hope, in the case suchproperties are confirmed in more articulated models, for thepossibility that well-defined tuning of realistic models couldallow for a successful representation of the baroclinic wavecomponent of the mid-latitude variability. Other componentsof atmospheric variability, in particular the low frequencyvariability, may, instead, be characterized by more criticalproperties, such as resonance, non-propagating behavior,etc.;

2. all of the statistical properties of the model are notonly smooth, but can be expressed as power-law scalingswith respect to TE . Therefore, it is possible to invertthe functional relationship for, e.g., the mean zonal wind〈U 〉, and parameterize, at least piecewise, all of the otherstatistical properties with respect to 〈U 〉. In general, theself-similarity could be of great help in setting up atheory for the overall statistical properties of the generalcirculation of the atmosphere and in guiding – on a heuristicbasis – both data analysis and realistic simulations, goingbeyond the unsatisfactory mean field theories and bruteforce approaches. A leading example for this would be the

possibility of estimating the sensitivity of the output ofthe system with respect to changes in the parameters. Theanalysis of scaling properties is emerging as new paradigmin the study of geophysical systems [69].

3. Self-similarity might result from statistical equilibration ofthe baroclinic conversion process, which provides a modern,statistical version of classical theories of the baroclinicjet. The observed equilibration somewhat resembles thedeterministic baroclinic adjustment envisaged in the ’70s,but takes place through more complex processes occurringin the strange attractor. Given the almost-linear nature ofthe wave-zonal flow interaction, it is possible to establisha simple closure of the properties of propagation andinstability of the waves with respect to a parameterizedzonal wind, which acts as integrator providing a time-varying index of refraction.

Three are the main, necessary directions of future work alongthe proposed line of approach to GAC:

1. Consolidating and explaining the knowledge (see [70])that in the real atmosphere the dynamics of travelingbaroclinic disturbances is dominated by the wave-zonal flowinteraction and the role of the wave–wave interaction isminor. Exploiting the potential of such a finding in modelingin detail the so called extratropical storm tracks [38] (seehttp://data.giss.nasa.gov/stormtracks/ for a nice tutorial),which are the everyday atmospheric manifestation of thebaroclinic dynamics we are debating.

2. Along the previous line, analyzing mathematically for thismodel the action of the zonal wind as an integrator of thewave disturbances. This can be accomplished by writingthe approximate evolution equations for the wave fields,indicated generically as Ex , in the form Ex = M(t)Ex , whereM is a stochastic matrix [18] written in terms of the U andm fields. The statistical properties of M may inform us onthe physical processes responsible for the wave dynamics.The generation and decay processes are quite relevant for themid-latitudes of the real atmosphere, because we have 2Dwavenumber–frequency spectral densities that only vaguelycorrespond, in a statistical sense, to well-defined dispersionrelations ω = ω(k), since at all frequencies the spectralwidth is relatively wide [21,22,53].

3. Analyzing all the debated properties in more articulatednumerical models. Some preliminary results in this sensehave been found by the authors and Vannitsem (unpublishedresults). An important technical issue in this context may bethe separation of the properties of the baroclinic mid-latitudejet system from the low frequency variability.

Acknowledgments

The authors wish to thank M. Felici, G. Gallavotti, V. Gupta,S. Lovejoy, P. Malguzzi, L. Smith, A. Tsonis, and S. Vannitsemfor scientific hints and encouragement, and the two reviewersfor relevant suggestions aimed at the improvement of the paper.V.L. wishes to thank ISAC-CNR for kind hospitality and the

http://data.giss.nasa.gov/stormtracks/


Centro di Cultura Scientifica A. Volta (Como, Italy) for partialfinancial support.

Appendix. Evolution equations of the single zonal wavetwo-layer model and numerical methods

As explained in Section 2, for the purposes of this studywe retain the zonally symmetric component plus only onewave component in the zonal direction of the fields φ(x, y, t)and τ(x, y, t). Since we intend to represent schematically thebaroclinic conversion processes, which in the real atmospheretake place on scales of the order of Lx/6 [21,22], we select thewave component having wave vector χ = 6 × 2π/Lx . We thenhave:

φ(x, y, t) = −

∫ y

π/2U (z, t)dz + A exp (iχx)+ c.c., (A.1)

τ(x, y, t) = −

∫ y

π/2m(z, t)dz + B exp (iχx)+ c.c.. (A.2)

By substituting (A.1) and (A.2) into equations (6) and (7) andprojecting onto the zonal Fourier modes of order n = 0 andn = 6, we obtain a set of partial differential equations for the6 real fields A1, A2, B1, B2, U , m, where A1 and A2 are thereal and imaginary parts of A, and similarly for B. Vanishingboundary conditions are chosen at y = 0, L y for all fields. Inthe case of U and m this amounts to setting no-flux boundaryconditions for the zonally symmetric component of φ and τ ,along the lines of Phillips [60]. A Fourier half-sine expansionof the fields is then carried out along y, with time-varyingcoefficients:

X =

JT∑j=1

X j sin(π j yL y

), (A.3)

where X represents each of the fields A1, A2, B1, B2, U , and m,and for the truncation order JT we have used the values JT = 8,16, 32, 64. With this definition, and because of (A.1) and (A.2),the latitudinal average of τ and φ vanishes in all cases, sothat the energy density (15) does not depend on the latitudinalaverage of τ , which has no physical relevance. By using theexpansion (A.3) and projecting on the basis functions sin(π j y

L y),

we then obtain a set of 6×JT ordinary differential equations forA1

j , A2j , B1

j , B2j , U j , m j . Particular care must be taken for the

nonlinear (quadratic) terms appearing in (6) and (7). Such termsresult into sums of quadratic products of the time-dependentvariables A1/2

j , B1/2j , U j , m j times y-dependent factors given

by quadratic products of the basis functions sin(π j yL y) and

their derivatives. We define a pseudospectral operator Π j (·)

for projecting such quadratic y-dependent terms onto the j-thbasis function sin π j y

L ythrough Fourier collocation: the factors

are evaluated pointwise at the JT equally spaced collocationpoints y1, . . . , yJT in the y-domain. Then, an inverse DiscreteSine Transform is carried out, yielding the Fourier coefficientsof the nonlinear terms, thus obtaining the projection. Thesoftware library fftw3, publicly available at www.fftw.org,has been adopted. The resulting system of ordinary differential

equations, which constitutes the base model of our study,is:

A1j =

1χ2 + w2

j

[−

2νE

H22(χ2

+ w2j )A

1j − χβA2

j

+2νE

H22(χ2

+ w2j )B

1j + Π j

(−χU A2

yy + χ3U A2

+ χUyy A2− χm B2

yy + χ3m B2+ χm yy B2

)], (A.4)

A2j =

1χ2 + w2

j

[−

2νE

H22(χ2

+ w2j )A

2j + χβA1

j

+2νE

H22(χ2

+ w2j )B

2j + Π j

(χU A1

yy − χ3U A1

− χUyy A1+ χm B1

yy − χ3m B1− χm yy B1

)], (A.5)

B1j =

1

χ2 + w2j +

2H2

2

[−

(2νE

H22

+2κH2

2

)(χ2

+ w2j )B

1j

−χβB2j +

2νE

H22(χ2

+ w2j )A

1j + Π j

(− χU B2

yy

+χ3U B2+ χUyy B2

+2

H22χU B2

− χm A2yy

+ χ3m A2+ χm yy A2

−2

H22χm A2

)], (A.6)

B2j =

1

χ2 + w2j +

2H2

2

[−

(2νE

H22

+2κH2

2

)(χ2

+ w2j )B

2j

+χβB1j +

2νE

H22(χ2

+ w2j )A

2j + Π j

(χU B1

yy

−χ3U B1− χUyy B1

−2

H22χU B1

+ χm A1yy

− χ3m A1− χm yy A1

+2

H22χm A1

)], (A.7)

U j = −2νE

H22(U j − m j )− 2χΠ j

×

(−A1 A2

yy + A2 A1yy − B1 B2

yy + B2 B1yy

), (A.8)

m j =1

2H2

2+ w2

j

[−w2

j2κH2

2m j + w2

j2νE

H22(U j − m j )

−2νN

H22(m j − m∗

j )+ Π j

(4χ

1H2

2(A1 B2

− A2 B1)yy

+ 2χ(−A1 B2yy + A2 B1

yy − B1 A2yy

− B2 A1yy)yy

)], (A.9)

where the explicit expression of the projected nonlinearterms is not presented here because of space limitation. Thenumerical solution of the system (A.4)–(A.9) is computed

http://www.fftw.org


with a standard Runge–Kutta–Fehlberg(4,5) algorithm [71]with adaptive step size, where the approximated solutionis carried by the order five method. The local truncationerror is kept below 1 · e − 6. The step size adjustmentprocedure is similar to that of DOPRI5, available at(www.unige.ch/˜hairer).

For the computation of the averages X , time series of315 360 adimensional time units (1000 years in natural units)have been computed for all values of JT , after discarding atransient of 5 years. The observables E(t), U (y, t), and m(y, t)have been sampled every 0.216 time units (four times a day),thereby obtaining time series of 1 460 000 elements.

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doi:10.1007/s00382-006-0213-x

doi:10.1007/s00382-006-0213-x

doi:10.1007/s00382-006-0213-x

doi:10.1007/s00382-006-0213-x

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doi:10.1146/annurev.earth.34.031405.125144








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